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We study gravitational lensing by compact objects in gravity theories thatcan be written in a Post-Post-Newtonian (PPN) framework: i.e., the metric isstatic and spherically symmetric, and can be written as a Taylor series in m/r,where m is the gravitational radius of the compact object. Working invariantly,we compute corrections to standard weak-deflection lensing observables at firstand second order in the ratio of the angular gravitational radius to theangular Einstein ring radius of the lens. We show that the first-ordercorrections to the total magnification and centroid position vanish universallyfor gravity theories that can be written in the PPN framework. This arises fromsome surprising, fundamental relations among the lensing observables in PPNgravity models. We derive these relations for the image positions,magnifications, and time delays. A deep consequence is that any violation ofthe universal relations would signal the need for a gravity model outside thePPN framework (provided that some basic assumptions hold). In practical terms,the relations will guide observational programs to test general relativity,modified gravity theories, and possibly the Cosmic Censorship conjecture. Weuse the new relations to identify lensing observables that are accessible tocurrent or near-future technology, and to find combinations of observables thatare most useful for probing the spacetime metric. We give explicit applicationsto the Galactic black hole, microlensing, and the binary pulsar J0737-3039.Comment: submitte

Formalism for testing theories of gravity using lensing by compact objects. II. Probing post-post-Newtonian metrics